Recently, epimorphisms between knot groups are studied by many researchers. In this talk, we will survey several (mainly the speaker's) results on this topic. Namely, all the pairs of prime knots with up to 11 crossings which admit meridional epimorphisms between knot groups are determined completely. Here a homomorphism is called meridional if it preserves meridians. On the other hand, we can construct a non-meridional epimorphism too. Finally, we consider the relationship between epimorphisms of two-bridge knot groups and their crossing numbers.